Question About Web IP Detection Tools

I needed to change my IP address so I used a web IP detection tool like whatismyip and wrote down my IP address. I tried unplugging the modem and router and restarting but when I went to whatismyip I still got the same IP address being detected. I contacted my ISP and told them that I wanted my IP address changed. They did a manual reset of the cable modem and told me to unplug the router. Once again, I still get the same IP address from any web IP detection tool.

Now, the guy at the ISP told me that I had a new IP address to the outside world and I should ignore the information from whatismyip because its only running a script. Did my IP address really change to the outside world? Should I ignore the IP address that gets detected by those web IP detection tools?

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